Beta-expansions, natural extensions and multiple tilings associated with Pisot units

Kalle, Charlene; Steiner, Wolfgang

doi:10.1090/S0002-9947-2012-05362-1

Beta-expansions, natural extensions and multiple tilings associated with Pisot units
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by Charlene Kalle and Wolfgang Steiner PDF

Trans. Amer. Math. Soc. 364 (2012), 2281-2318 Request permission

Abstract:

From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\beta$ and the greedy $\beta$-transformation. In this paper, we consider different transformations generating expansions in base $\beta$, including cases where the associated subshift is not sofic. Under certain mild conditions, we show that they give multiple tilings. We also give a necessary and sufficient condition for the tiling property, generalizing the weak finiteness property (W) for greedy $\beta$-expansions. Remarkably, the symmetric $\beta$-transformation does not satisfy this condition when $\beta$ is the smallest Pisot number or the Tribonacci number. This means that the Pisot conjecture on tilings cannot be extended to the symmetric $\beta$-transformation.

Closely related to these (multiple) tilings are natural extensions of the transformations, which have many nice properties: they are invariant under the Lebesgue measure; under certain conditions, they provide Markov partitions of the torus; they characterize the numbers with purely periodic expansion, and they allow determining any digit in an expansion without knowing the other digits.

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